The present invention relates to improved statistical methods making use of estimated or approximated data and, more particularly, to the study of subject growth rates and the statistical treatment of subject growth data, particularly infant size data such as weight, length and head circumference.
Data is collected and analyzed to determine the effects of various influences on growth, particularly infant growth. Such influences may include genetic factors, environmental factors or interventions such as nutritional or medical treatment For example, many studies have been conducted to determine how differences in infant formula composition affect growth of an infant. Data must generally be interpreted using statistical methods to separate and distinguish xe2x80x9capparentxe2x80x9d effects that are due to random, uncontrolled variability from xe2x80x9ctruexe2x80x9d effects that result from differences in the formulas tested in the study.
One source of uncontrolled variability that influences the precision and reliability of growth data is the timing at which growth measurements are made. Typically, a researcher wants to compare study outcomes (e.g. size parameters such as weight, length and head circumference, or mental development parameters, such as Bayley""s) as a function of time, to determine what relationship, if any, exists between the factor or intervention being tested and the subject""s growth rate. Study outcomes, such as subject sizes, would ideally be measured at predetermined ages selected by the researcher, for example at precisely 2, 4 and 6 months. In practice however, the outcome measurements may not be made at precisely the targeted time. For example, in infant growth studies, the infant-subjects often are not brought in for measurements at the precise time predetermined by the researcher, so subject growth data is collected at irregular times. Because infants grow and change very quickly, even a few days difference between the predetermined target time for a measurement and the actual time at which the measurement is made is a source of uncontrolled variability that can significantly affect the interpretation of the data. This variability can reduce the usefulness of the data and reduce the precision of the statistical comparisons.
To address this problem of time-variability of measurements in infants, it has been suggested that an infant growth model be used. A number of growth models have been proposed in the literature. Count, E., 15 Human Biology 1-32 (1943) discloses a size modeling function, including one of the form ƒa,b,c(t)=a+b t+c log(t+1). Guo, et al., 119 J. Pediatr. 334-362 (1991) describe a function of the form ƒa,b,c(t)=a+b log(t+1)+c sqrt(t+1). Karlberg et al., 48 (Suppl. 1) European Journal of Clinical Nutrition S25-S44 (1994) teach a model of the form ƒa,b,c(t)=a+b (1xe2x88x92exp(xe2x88x92cx)). These and other infant growth models are reviewed by Peerson, et al., in an article titled Use of Growth Models to Describe Patterns of Length, Weight, and Head Circumference among Breast-Fed and Formula-Fed Infants: The Darling Study, Human Biology, 65(4):611-626, 1993. These known infant growth models have limited usefulness, however, and are generally inapplicable to preterm infant growth data.
In addition, a Gompertz function has been applied to growth modeling in the adolescent by Pasternack and Shohoji in Essays in Probability and Statistics (Ikeda, Sadao, et. al. eds), Fitting a Gompertz Curve to Adolescent Standing Height Growth Data, (Chapter 35, pp. 559-577, Shinko Tsusho, Tokyo, 1976), and by Deming, Human Biology, 29:83-122 (1957). A Gompertz function has also been used to model growth of whole organisms, both pre- and post-natal, as well as various organs and parts of whole organisms in a series of 1960""s papers by A. K. Laird. See, for example, Laird, Dynamic of Relative Growth, Growth 29, 249-363 (1965); Laird, Postnatal growth of birds and mammals, Growth 30:349-363 (1966); and Laird, Evolution of the human growth curve, Growth, 31:345-355 (1967).
A portion of applicants"" own work was published in abstract form: Zhang et al, Use of a Gompertz curve to describe patterns of early growth in term and preterm infants (Abstr. #165) Amer. J. of Human Biology, 10:1 pp 139-140, 1998. To applicants"" knowledge, however, the use of such Gompertz functions has not previously been applied to evaluate studies in which time is the independent variable and for which the data may not conform precisely to the desired times. To applicants"" knowledge, the function has not previously been used to predict or approximate data points for a time common to multiple subjects, followed by the comparison of such estimated data for evaluation of the intervention of the clinical trial. Weissfeld and Kshirsagar, Austral. J. Statist., 34(2):161-168 (1992) describe a modified use of a growth modeling function (not Gompertz) to adapt it to a same-patient-multiple-treatments format in order to test certain hypotheses about the treatments.
In one aspect, the invention provides a method of interpreting results from a study, said method comprising the steps of:
obtaining measured size data for each of two or more subjects regarding the subjects"" sizes as a function of an independent variable, t, that corresponds to a measure of the subjects"" ages, wherein t for at least one subject differs from t for at least one other subject;
determining for each of the subjects a set of values for the three parameters a, b, and c of a three parameter size modeling function defined by ƒa,b,c(t)=a exp (b(1xe2x88x92exp(xe2x88x92t))), to provide an optimal fit of the size modeling function to the data regarding the subject""s size as a function of t;
estimating for each of the subjects, using said size modeling function and said set of determined values for the parameters, sizes for the subject at a particular age, which particular age is the same for all of the subjects; and
comparing the estimated sizes to interpret the study results.
Preferably the optimal fit is obtained by minimizing the least squares error function. The age variable may be chronological age and, in an infant growth studyxe2x80x94especially a preterm infant growth study, the age variable is preferably gestation-adjusted age. Comparing is a broad term that encompasses both simple comparisons and more complicated statistical analyses.
In another aspect, the invention provides a method of conducting a study, wherein the results are interpreted in accordance with the method described above.
In yet another aspect, the invention provides a device comprising:
memory means encoded with or adapted to receive instructions, said instructions capable of directing a computer provided with measured outcome data regarding a subject size as a function of an age, t, to calculate values for three parameters, a, b, and, c of a three parameter subject size modeling function defined by ƒa,b,c(t)=a exp (b(1xe2x88x92exp(xe2x88x92ct))); such that the size modeling function with the calculated parameter values gives an optimal fit of the function to the data;
instructions to use the calculated values for parameters a, b, and c in the size modeling function to generate estimated data for each subject at a common predetermined age that is different from t for at least one subject; and
output means for presenting the estimated data.
In a preferred embodiment, the optimal fit is a least squares fit.
Another aspect of the invention provides a method of processing data for improved interpretation, said method comprising the steps of:
obtaining measured outcome data for a plurality of test subjects at times, ti, corresponding to a measure of time, wherein ti for at least one subject differs from ti for at least one other subject;
determining for each subject a set of values for the three parameters a, b, and c of a three parameter modeling function defined by ƒa,b,c(t)=a exp (b(1xe2x88x92exp(xe2x88x92ct))) that relate outcome data to an independent time variable t, to provide an optimal fit of the modeling function to the measured outcome data for each subject over all ti;
estimating outcome data for at least one of the subjects, using said modeling function and said the determined set of parameter values, for a particular time that is different from ti at which the outcome data was measured; and
comparing the estimated outcome data from the at least one subject with estimated or measured outcome data from at least one other subject to interpret the results.
In this aspect, it is preferable that the outcome data be a measure of growth, including but not limited to anthropometric growth outcomes like weight, stature/length or head circumference. It is also preferable that the estimated outcome data be estimated for multiple, if not all the subjects; that the independent time variable is gestation-adjusted age; and that a computer processing means is used for some or all of the steps.
In a further aspect, the invention provides a process applied to data regarding a measure of subject size as a function of an independent variable t, corresponding to subject age, comprising the steps of: entering the data into a computer programmed to calculate values for three parameters, a, b, and, c in a three parameter subject size modeling function defined by ƒa,b,c(t)=a exp (b(1xe2x88x92exp(xe2x88x92ct))), wherein the computer is programmed to calculate the parameter values such that the modeling function with the calculated parameter values gives an optimal fit of the function to the data.
In a further aspect, the invention provides a method applied to data regarding a measure of subject size as a function of an independent variable corresponding to age, t, comprising the steps of: selecting a three parameter size modeling function defined by ƒa,b,c(t)=a exp (b(1xe2x88x92exp(xe2x88x92ct))); determining values for the three parameters a, b, and c to provide an optimal fit of the size modeling function to the subject size data; and recording the values for the three parameters, a, b, and c.
In each of the aspects described above, preferred embodiments include the following features: the optimal fit is a least squares fit; the parameter values are determined with the aid of a computer or similar processing instrument; and the estimated data is recorded on computer readable media. Further, it is often preferable that the subject size outcome is selected from the group consisting of weight, length, and head circumference; and the measure of age is gestation-adjusted age.